Lahenda frac{sqrt{5}div[sqrt{8}+sqrt{5}]}{sqrt{8}-sqrt{5}}

$$\frac { \sqrt { 5 } \div [ \sqrt { 8 } + \sqrt { 5 } ] } { \sqrt { 8 } - \sqrt { 5 } }$$

Arvuta

$$\frac{\sqrt{5}}{3}\approx 0,745355992$$

Lahenduse etapid

Tegurda 8=2^{2}\times 2. Kirjutage \sqrt{2^{2}\times 2} toote juured, kui see ruut \sqrt{2^{2}}\sqrt{2}. Leidke 2^{2} ruutjuur.

$$\frac{\frac{\sqrt{5}}{2\sqrt{2}+\sqrt{5}}}{\sqrt{8}-\sqrt{5}}$$

Ratsionaliseerige korrutades lugeja ja 2\sqrt{2}-\sqrt{5} nimetaja \frac{\sqrt{5}}{2\sqrt{2}+\sqrt{5}} nimetaja.

$$\frac{\frac{\sqrt{5}\left(2\sqrt{2}-\sqrt{5}\right)}{\left(2\sqrt{2}+\sqrt{5}\right)\left(2\sqrt{2}-\sqrt{5}\right)}}{\sqrt{8}-\sqrt{5}}$$

Mõelge valemile \left(2\sqrt{2}+\sqrt{5}\right)\left(2\sqrt{2}-\sqrt{5}\right). Korrutustehte saab ruutude vaheks teisendada järgmise reegli abil: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\frac{\sqrt{5}\left(2\sqrt{2}-\sqrt{5}\right)}{\left(2\sqrt{2}\right)^{2}-\left(\sqrt{5}\right)^{2}}}{\sqrt{8}-\sqrt{5}}$$

Laiendage \left(2\sqrt{2}\right)^{2}.

$$\frac{\frac{\sqrt{5}\left(2\sqrt{2}-\sqrt{5}\right)}{2^{2}\left(\sqrt{2}\right)^{2}-\left(\sqrt{5}\right)^{2}}}{\sqrt{8}-\sqrt{5}}$$

Arvutage 2 aste 2 ja leidke 4.

$$\frac{\frac{\sqrt{5}\left(2\sqrt{2}-\sqrt{5}\right)}{4\left(\sqrt{2}\right)^{2}-\left(\sqrt{5}\right)^{2}}}{\sqrt{8}-\sqrt{5}}$$

\sqrt{2} ruut on 2.

$$\frac{\frac{\sqrt{5}\left(2\sqrt{2}-\sqrt{5}\right)}{4\times 2-\left(\sqrt{5}\right)^{2}}}{\sqrt{8}-\sqrt{5}}$$

Korrutage 4 ja 2, et leida 8.

$$\frac{\frac{\sqrt{5}\left(2\sqrt{2}-\sqrt{5}\right)}{8-\left(\sqrt{5}\right)^{2}}}{\sqrt{8}-\sqrt{5}}$$

\sqrt{5} ruut on 5.

$$\frac{\frac{\sqrt{5}\left(2\sqrt{2}-\sqrt{5}\right)}{8-5}}{\sqrt{8}-\sqrt{5}}$$

Lahutage 5 väärtusest 8, et leida 3.

$$\frac{\frac{\sqrt{5}\left(2\sqrt{2}-\sqrt{5}\right)}{3}}{\sqrt{8}-\sqrt{5}}$$

Tegurda 8=2^{2}\times 2. Kirjutage \sqrt{2^{2}\times 2} toote juured, kui see ruut \sqrt{2^{2}}\sqrt{2}. Leidke 2^{2} ruutjuur.

$$\frac{\frac{\sqrt{5}\left(2\sqrt{2}-\sqrt{5}\right)}{3}}{2\sqrt{2}-\sqrt{5}}$$

Avaldage \frac{\frac{\sqrt{5}\left(2\sqrt{2}-\sqrt{5}\right)}{3}}{2\sqrt{2}-\sqrt{5}} ühe murdarvuna.

$$\frac{\sqrt{5}\left(2\sqrt{2}-\sqrt{5}\right)}{3\left(2\sqrt{2}-\sqrt{5}\right)}$$

Taandage -\sqrt{5}+2\sqrt{2} nii lugejas kui ka nimetajas.

$$\frac{\sqrt{5}}{3}$$